#627845

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

(2,5)(7,8)(11,15)(12,16)(18,21)(22,24)(25,26,29,28)(27,31,32,30);(1,2,6,5)(3,8,4,7)(9,10,14,13)(11,12,15,16)(27,32)(30,31);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32);(1,4,6,3)(2,8,5,7)(9,12,14,16)(10,15,13,11);(1,5,6,2)(3,8,4,7)(9,13,14,10)(11,16,15,12)(17,19,20,23)(18,22,21,24)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=f,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 535041);